Simplify the following expression and state the condition under which the simplification is valid: $n = \dfrac{x^2 + 13x + 40}{x^2 + 9x + 8}$
Answer: First factor the expressions in the numerator and denominator. $ \dfrac{x^2 + 13x + 40}{x^2 + 9x + 8} = \dfrac{(x + 5)(x + 8)}{(x + 1)(x + 8)} $ Notice that the term $(x + 8)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(x + 8)$ gives: $n = \dfrac{x + 5}{x + 1}$ Since we divided by $(x + 8)$, $x \neq -8$. $n = \dfrac{x + 5}{x + 1}; \space x \neq -8$